www.sciencemag.org/content/351/6272/482/suppl/DC1

Supplementary Materials for Ancient Babylonian astronomers calculated Jupiter’s position from the area

under a time-velocity graph

Mathieu Ossendrijver*

*Corresponding author. E-mail: mathieu.ossendrijver@hu-berlin.de

Published 29 January 2016, Science 351, 482 (2016) DOI: 10.1126/science.aad8085

This PDF file includes:

Materials and Methods Figs. S1 to S4 References

Materials and Methods

Transliterations, translations and photographs of the cuneiform tablets

Cuneiform Texts A-E were transliterated and translated from the original clay tabletsduring several visits to the Study Room of the Middle Eastern department of the BritishMuseum (London) between 2002 and 2015 and from photographs made on theseoccasions. All tablets and fragments are identified by registration numbers assigned bythe British Museum (BM). The transliterations were made in accordance withAssyriological conventions, i.e. logograms are written in capitals, Akkadian phoneticsigns in italics, [x] indicates a sign broken away, [...] a break of unknown length, and x⸢ ⸣a damaged sign. In the translations, missing text is indicated by [...], untranslatable textby .... Note that the cuneiform notation for sexagesimal numbers lacks an equivalent ofour decimal point. In the transliterations this feature is maintained by separating all digitsby a period (.). In the translations the absolute value of each number, as inferred from thecontext, is indicated by separating the digit pertaining to 600 from the next one pertainingto 60–1 by a semicolon (;) and all other digits by commas. Also note that initial and finalvanishing digits are not written in cuneiform. For instance, in line 1 of Text A the numberwritten as 12 represents 0;12, which stands for 12/60, 1 represents 1,0 = 60, and 9.30represents 0;9,30 = 9/60+30/602. All displacements of Jupiter in Texts A-E are measuredin degrees (º), but this unit is, as usual in Babylonian mathematical astronomy, notmentioned explicitly.

Provenance and date of Texts A-E

The tablets on which Texts A-E are written were excavated unscientifically in Iraq in the19th century, along with thousands of other tablets (16). Even though their exact findspotis not documented, there is a consensus that the astronomical tablets from theseexcavations originate from Babylon, the main center of Babylonian astronomy during thefirst Millennium BCE (2, 16). This is confirmed by occasional references to Babylon'smain temple on some of the astronomical tablets (1, 2).

Due to a combination of factors it is impossible to assign very precise dates to Texts A-E.First, they lack a colophon that might have mentioned a date of writing or the name of adatable astronomer. Second, they do not mention datable astronomical phenomena. Third,since they were excavated unscientifically no information about the archaeologicalstratigraphy of the tablets is available. However, the range of possible dates is constrainedby the following considerations. First, the computations in Texts A-E employ eclipticalcoordinates, which implies that they were written after the estimated date of introductionof these coordinates near the end of the fifth century BCE (3). Second, many of the othertablets from Babylon that deal with mathematical astronomy have been dated. The datesof these tablets roughly extend from 350 BCE to 50 BCE, while their distribution peaksbetween 180 BCE and 100 BCE (1). Texts A-E therefore very likely date from 350-50BCE, the range of most probable dates being 180-100 BCE.

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Text A

Text A is inscribed, in a highly cursive hand, on the tablet BM 40054, which measures 4.3x 3.8 x 2.0 cm (Fig. 1). The tablet arrived in the British Museum in 1881, having beenexcavated unscientifically in Iraq, very likely in Babylon (16). The tablet is nearly intact,apart from a slice of clay, perhaps corresponding to one line of text, that is missing fromthe lower edge of the obverse and the upper edge of the reverse. The surface of the tabletis notably curved. The tablet contains a single procedure concerning scheme X.S1 forJupiter. It was previously unpublished. Text A comprises lines 1-7 of the obverse; theremaining approximately 12 lines deal with subsequent intervals of Jupiter's synodiccycle and will be published elsewhere.

Transliteration

1 ME IGI 12 EN 1 ME 9.30

2 12 u3 9.30 21.30 A 30

3 10.45 A 1 10.4 5⸢ ⸣

4 TA 1 ME GI EN 1 ME 1.30⸢ ⸣

5 9.30 u3 1.30 11 A.RA2 30 5.30

6 5.30 A 1 5.30 (erasure) 10.45 u3 5.30

7 16 .15 PAP.PAP TA IGI EN U⸢ ⸣ Š 16.15 DU.DU?

Translation

1 The day when it appears: 0;12, until 1,0 days, 0;9,30.

2 0;12 and 0;9,30 is 0;21,30, times 0;30

3 is 0;10,45, times 1,0 is 10;45.

4 After completing 1,0 days, until 1,0 days 0;1,30.

5 0;9,30 and 0;1,30 is 0;11, times 0;30 is 0;5,30.

6 0;5,30 times 1,0 days is 5;30. (erasure) 10;45 and 5;30 is

7 16;15, the total. From appearance until station the motion is 16;15.

Philological remarks

2, 3, 6) In these lines the usual logogram A.RA2, „times“, is abbreviated to A.

6) After 5.30 there is an erased sign that was overwritten by the 10.

7) DU.DU?: the sign following DU is written on the edge in a crammed manner. It lookslike BI with a horizontal wedge below it. This sign is here assumed to be a deformed DU.

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Like DU alone, the logogram DU.DU? probably represents alāku, „course; motion“, or arelated noun derived from this verb. Alternatively it may represent the present tense ofthis verb, illak, „it proceeds“.

Commentary

Of the five currently known tablets and fragments concerned with Jupiter's scheme X.S1,this is the best preserved one. However, compared to the other tablets its formulation ismore terse. The name of Jupiter is not mentioned and the significance of severalparameters, e.g. 0;12 and 0;9,30, is not indicated. Two close duplicates of Text A, BM36801 and BM 41043, published as Nos. 21 and 22 in (2), clarify the meaning of Text A.They do mention the name of Jupiter and they qualify 0;12 as a value of its dailydisplacement (along the ecliptic), as can be seen in lines 1-3 of BM 36801:

Transliteration

1 MUL2.BABBAR ME IGI 12 ZI-šu2 EN [... A.RA⸢ ⸣ 2 ]

2 1 ME DU-ma 10.45 TA 1 ME GI [...] ⸢ ⸣

3 5.30 5.30 A.RA2 1 ME DU-⸢ma [5.30 ...]⸣

Translation

1 Jupiter. The day when it appears its (daily) displacement is 0;12, until [... times]

2 1,0 days you multiply, it is 10;45. After completing 1,0 days [...]

3 0;5,30. You multiply 0;5,30 by 1,0 days, it is [5;30 ...]

This agreement between Text A and BM 36801 proves that Text A pertains to Jupiter andthat the numbers 0;12 and 0;9,30 are values of its daily displacement. That this motionproceeds along the ecliptic and that the displacements are measured in degrees followsfrom a comparison with other tablets about planetary motion (1, 2).

General remarks about Texts B-E

Texts B-D and, as was argued, Text E, deal with the same trapezoid with short side0;9,30, long side 0;12 and width 1,0, but they are not exact duplicates. That is, in each ofthe four texts the trapezoid procedure is formulated in slightly different words. Severalphrases appear in more than one text, but not always in the same order. Text E deviatesthe most, being the only one that partly preserves the end of part II.

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Text B

Text B is written on the fragment BM 34757, which measures 5.1 x 6.3 x 2.0-2.9 cm (Fig.S1). It arrived in the British Museum around 1879, having been excavatedunscientifically in Iraq, most likely in Babylon (16). The fragment belongs to the left halfof a tablet. Apart from the left edge, no other edges of the original tablet are preserved.The textual restorations imply that about 2 cm of clay are missing from the right side;hence the original width was about 7 cm. Text B occupies one side of BM 34757, whichmight equally be the obverse or the reverse. The other side partly preserves three moreprocedures about Jupiter. For an edition of the complete tablet see No. 38 in (2). Thepresent edition incorporates several significant corrections to the previous one (2).

Transliteration

1 xx MUL⸢ 2 .BABBAR ⸣ ina šu-tam-ḫi-⸢ri [12 SAG]⸣

2 GAL-tu2 9.30 SAG TUR-tu2 UŠ-šu2 E[N.NAM]

3 10.45 ana UGU KI IGI TAB-ma ina 1-šu [xx]

4 DU3.DU3.BI ša2 SAG.KI GU4 SAG-⸢su [xx]⸣

5 ana muḫ-ḫi a-ḫa-miš2 TAB-ma 1/2-šu2 GIŠ-⸢ma [A.RA⸣ 2 1-šu DU-ma]

6 10.45 A.ŠA3 10.45 ana UGU KI [IGI TAB xx]

7 s ṣal-pi ina KASKAL-šu2 ina ŠA3-šu2 (erasure) SE3.AM3 ⸢u4 -[⸣ mu ša2]

8 ana s ṣal-pi DIB-iq tam-mar ki-ma-a ⸢mu -[⸣ šam-ša2 x]

9 10.45 šu-u2 tam-mar ki BAR šu-u2 12 SAG [GAL-⸢ ⸣ tu2 A.RA2 12 DU-ma]

10 2.24 9.30 SAG t ṣe-ri-tu2 A.RA2 9.3[0 DU-ma 1.30.15]

11 TA 2.24 ZI-ma ša2 re-ḫi A.RA2 KI [xxx]

12 ⸢ša2? 10.4 5 DU-⸣ ma 26.52.30 9.30 xx [xx]⸢ ⸣

(unknown number of lines missing)

Translation

1 ... of Jupiter by squaring. [0;12 is]

2 the large [side], 0;9,30 is the small side, w[hat] is its station?

3 You add 10;45 to the position of appearance, and in sixty [days ...]

4 Its procedure: the [...] side(s) of the trapezoid

5 you add together, you compute half of it and [you multiply it by sixty days, it is]

6 10;45, the area; 10;45 [you add] to the position [of appearance. ...]

7 You place the crossing in its path, in its middle. The da[y when](or: You se[e] the crossing in its path, whereby it is halved. [The day when])

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8 it passes the crossing, you see how long it is spen[ding ...]

9 (Concerning) this 10;45, you see when (or: how; that) it is halved: 0;12, the [large]side, [you multiply by 0;12, it is]

10 0;2,24. 0;9,30, the pinched side, [you multiply] by 0;9,30, [it is 0;1,30,15.]

11 You subtract it from 0;2,24, what remains you multiply by ... [...]

12 of? 10;45, it is 0;0,26,52,30. 0;9,30 ... [...]

(unknown number of lines missing)

Philological remarks

1) The first two signs are too strongly damaged to be identified. At the end of the line,traces compatible with the beginning of ri are visible. šu-tam- i-ḫ ⸢ri : ⸣ šutam iruḫ is mostlikely a variant spelling of šutam uruḫ , infinitive of the Št stem of ma āruḫ , which means„to square“ in mathematical contexts (4). Apart from the present instance this verb is notattested in the astronomical corpus. Because several numbers are squared later on in theprocedure, „squaring“ might be the appropriate translation. However, since the phraseoccurs at the beginning of the procedure it may also be interpreted as a more generalreference to the geometrical method underlying the entire procedure.

2) UŠ-šu2: in the present context this most likely represents nēmettašu, „its station“, i.e.Jupiter's first station. The traces that remain of the last sign are compatible withE[N.NAM] = mīnû, „what“, as proposed in (1).

3) 1-šu = uššu, „sixty“: here the number 60 is written semi-phonetically and not insexagesimal place value notation, perhaps in order to avoid a misinterpretation, since inthat notation 1 and 60 would both be rendered as 1.

7) s ṣal-pi: the technical term s ṣalpu is tentatively translated as „crossing“ (2). After ŠA3-šu2

one or two signs were partly erased, most likely on purpose by the scribe. The survivingtraces suggest that s ṣal-pi, „crossing“, was previously written there. SE3: in the presentcontext, two readings of this logogram appear possible. First, a form of mašālu, „to behalf; equal“, most likely a stative of the G stem, mašil, „it is halved“. The pseudo-Sumerian affirmative suffix AM3 can be interpreted as a marker of this stative form. Inthis reading of SE3 the preceding ina libbi(ŠA3)-šu2 is most suitably interpreted as theinstrumental conjunction „whereby“ (20). A second plausible option is to interpret SE3 asa second person of the present tense of the G stem of šakānu, „to place“, i.e. tašakkan,„you place“, in which case ina libbi(ŠA3)-šu2 can be translated „within it; in its middle“.Neither interpretation can be ruled out.

8) tam-mar ki-ma-a ⸢mu -[⸣ šam-ša2 x], „you see how long it is spend[ing ...]“: as in line 9,the phrase following tammar, „you see“, is best interpreted as an indirect question (17-19). For a discussion of mušamšâ see the commentary to Text D, line 6.

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9) 10.45 šū tammar kī mašlu?(BAR) šū, „(Concerning) this 10;45: you see when it ishalved“. Since 10.45 šū, „this 10;45“, is in the nominative case, it is not the object of thefollowing tammar, „you see“. Instead, the object of tammar, „you see“, is theimmediately following clause introduced by the conjunction kī, „when; how; that“ (17-19). In the present context, the most suitable interpretation of this clause is an indirectquestion („when; how“). Alternatively it can be an object clause („that“), but they areusually introduced by ša (18). The subject of the clause is šū, „that; this; it“, the predicateBAR, which most likely represents a form of mašālu, „to be half; equal“. It is hereassumed to represent mašlu, a subjunctive of the stative form of the G stem, „it ishalved“, but some other form of this verb would also be possible.

12) ⸢ša2? 10.4 5, ⸣ „of 10;45“: the meaning of this incompletely preserved phrase is notfully clear. It presumably belongs to a descriptive term that qualifies the factor 0;30(=1/2) by which the outcome of line 11 is multiplied here (see the commentary below).

Commentary

Text B preserves 12 nearly complete lines of a trapezoid procedure. Line 1 is probablythe actual first line of the procedure. Lines 1-6 belong to part I; lines 7-12 to part II,which continued beyond line 12 for an unknown number of lines.

Part I: in lines 1-2 two measures of the trapezoid are preserved. The problem to be solvedis formulated as a question for the station of Jupiter (line 2). In line 3 the number 10;45 isadded to the position of Jupiter at its first appearance, i.e. λ60=λ0+10;45º. Thecorresponding time interval of 60 days is also preserved. In the following lines thisnumber, 10;45, is computed as the area of the trapezoid (lines 4-6). Its addition toJupiter's position at the first appearance is then repeated, i.e. λ60=λ0+10;45º (line 6).

Part II begins with a statement concerning the „crossing“ (line 7), which is said to belocated in the middle of Jupiter's path. This is followed by a statement of the quantity thatis asked for, namely the time in which Jupiter reaches the „crossing“ (lines 7-8). Thesolution procedure is partly preserved in lines 9-12. The quantity asked for is herespecified as the time in which Jupiter covers half the distance 10;45 (line 9). The solutionprocedure begins with a computation of v0

2 (line 9) and of v602 (line 10). The latter is

subtracted from the former, v02–v60

2, which is multiplied by 0;30 (=1/2), leading to(0;2,24–0;1,30,15)/2 = 0;0,26,52,30 = u2.

Text C

Text C is written on BM 34081+34622+34846+42816+45851+46135, which measures22.6 x 13.1 cm. The six known fragments of this tablet arrived in the British Museumbetween 1878 and 1881, having been excavated unscientifically in Iraq, most likely inBabylon (16). The left and right edges of the original tablet are partly preserved, butnothing remains of the original upper and lower edges. On both sides the tablet is dividedinto three columns. It is inscribed with a compendium of at least 32 procedures for

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Jupiter. Text C is the fifth procedure, which occupies lines 20-24 of column i on theobverse (Fig. S2). For an edition and photograph of the complete tablet see No. 18 in (2).The present edition incorporates several corrections to the previous one (2).

Transliteration

20 [xxxxxxx] 1-en SAG.KI GU4 ša2 12 SAG

21 [GAL-tu2 9.30 SAG TUR-tu2 xx] 1-⸢ šu ME 10.45 A.ŠA⸣ 3-šu2

22 [xxxxxxxx] ME 10.45 KI DU⸢ ⸣

23 [xxxxxxxxx] x ⸢ ⸣ t ṣe-ri-tu2

24 [xxxxxxxxxxx] x⸢ ⸣

[unknown number of lines missing]

Translation

20 [...] one trapezoid of which the [large] side is 0;12,

21 [... the small side is 0;9,30 ... times] sixty days is 10;45, its area.

22 [... In sixty] days the position proceeds 10;45.

23 [...] pinched ...

24 [...] ...

[unknown number of lines missing]

Philological remarks

22) The new reading qaqqaru(KI) illak(DU), „the position proceeds“, replaces E11, „yousubtract“ (2).

Commentary

Text C preserves up to 9 final signs of the first 5 lines of a trapezoid procedure. In whatremains of part I (lines 20-22), two measures of the trapezoid are provided (lines 20-21)and its area is obtained as 10;45 (line 21). In line 22 this is declared to be the distance bywhich the position (of Jupiter) proceeds, i.e. λ60=λ0+10;45º. What remains of lines 23-24is insufficient for identifying their meaning, but they probably belong to part II, whichcontinued below line 24 for an unknown number of lines.

Text D

Text D is inscribed on the previously unpublished fragment BM 35915, which measures3.5 x 4.0 x 0.8 cm (Fig. S3). It arrived in the British Museum around 1880, having beenexcavated unscientifically in Iraq, most likely in Babylon (16). No edges of the original

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tablet are preserved. Text D occupies one entire side of the fragment; the other side isdestroyed.

Transliteration

[unknown number of lines missing]

1 [...] xxxx [...]⸢ ⸣

2 [... 10].45 ana UGU KI I[GI TAB-ma ...]

3 [...] ⸢ki -⸣ ma-a ana s ṣal-pi DIB [...]

4 [... 10.45 šu]-⸢u2 ⸣ tam-mar ki-ma BAR-šu2 [...]

5 [... 9.30] SAG.KI t ṣe-ri-tu2 A. RA⸢ 2 [9.30 DU-⸣ ma 1.30.15 ...]

6 [...] DU? ⸢ ⸣ ki-ma mu-šam-ša2 x [...]⸢ ⸣

7 [...] 1 .30. 1 5 ⸢ ⸣ ⸢ ⸣ ana UGU x [...]⸢ ⸣

8 [...] DU?⸢ ⸣ al-la DIB? [...]⸢ ⸣

9 [...] x ⸢ ana ⸣ s ṣal-pi [...]

[unknown number of lines missing]

Translation

[unknown number of lines missing]

1 [...] ... [...]

2 [... 10];45 [you add] to the position of appea[rance, and ...]

3 [...] How long does it pass to the crossing [...]

4 [... Concerning th]is [10;45], you see when (or: how; that) it is halved: [...]

5 [... You multiply 0;9,30], the pinched side, times [0;9,30, it is 0;1,30,15 ...]

6 [...] You multiply [... times ...] ... how long it is spending ... [...]

7 [... You add] 0;1,30,15 to ... [...]

8 [...] ... beyond ... [...]

9 [...] ... to the crossing [...]

[unknown number of lines missing]

Philological remarks

4) For this clause see Text B, line 9. BAR can be read with a form of the verb mašālu, „tohalve; be equal“, but it might also be read „half; 1/2“, in which case one can translate „...you see how its half [...]“.

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6) mu-šam-ša2: the most plausible interpretation is mušamšâ, a participle of šumšû,literally „to spend the night“, which presumably has the more general meaning „to spendtime; stay“ here. See also the philological remarks to line 8 of Text B.

Commentary

Text D preserves portions of 9 lines of a trapezoid procedure. Lines 1-2 belong to part I;lines 3-9 to part II. Line 1 was most likely preceded by several more lines and part IIcontinued below line 9 for an unknown number of lines. The name of a planet is notpreserved, but a comparison with Texts A-C strongly suggests that Text D deals withJupiter.

In what remains of part I, 10;45, Jupiter's total displacement after 60 days, is added to itsposition at first appearance, i.e. λ60=λ0+10;45º (line 2). The problem to be solved in partII, namely to compute the time in which a planet, most likely Jupiter, covers half thedistance 10;45, is partly preserved in line 3. The solution procedure is introduced in line 4and then executed. Line 5 preserves the computation of v60

2. The computation ofv0

2=0;2,24 is missing, but a comparison with Text B (lines 9-10) suggests that it waswritten at the end of line 4. In line 7 v60

2 is added to something, presumably v02. Lines 8-9

are too damaged for an interpretation.

Text E

Text E is inscribed on the tablet BM 82824+99697+99742, which measures 8.3 x 5.3 x1.2 cm. The three fragments arrived in the British Museum in 1883 and 1884, havingbeen excavated unscientifically in Iraq, most likely in Babylon (16). The fragments donot include any portion of the original edges of the tablet. Only one side is inscribed; theother side is destroyed. Text E is the first preserved procedure (Fig. S4). It is followed bytwo more procedures, both concerned with Jupiter. The textual restorations in the latterimply that not much clay is missing from the left side and perhaps a few cm from theright side. For an edition and a photograph of the complete tablet see No. 40 in (2). Thepresent edition incorporates numerous corrections to (2).

Transliteration

[unknown number of lines missing]

1 [...xxxxxxxxx] xxxx [...]⸢ ⸣

2 [...] UŠ? xxx [xxxx] x UŠ 12? SAG.KI G[AL-⸢ ⸣ ⸢ ⸣ tu4 ...]

3 [...] x SAG.KI TUR-⸢ ⸣ tu4 E? x [xxx] 30? ⸢ ⸣ ⸢ uz -⸣ zab-bil IGI 2.50? [...]

4 [...] TA U4.MEŠ.AM3 KI UD x ⸢ u4-mu⸣ ša2 KI ⸢s ṣal⸣-pi DIB-qa tam-mar 3 2 [...]⸢ ⸣

5 [...] re-ḫi 27? 28 KI U⸢ 4 -⸣ ka TAB-ma u4-me KI⸢ s ṣal-pi DIB-⸣ qa tam-m[ar ...]

6 [...] ⸢ki -⸣ i TAG4 9. 30 ⸢ ⸣ ana UGU 10.50 TAB-ma 1/2-šu2 GIŠ-ma A.RA2 32 U⸢ 4 .⸣

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[MEŠ DU-ma 5.25.20 ...]

7 [...] xx BU.AM⸢ 3 10⸣ tu!-šaḫ-ḫi-zu PI AN NU TUKU 50 ⸢ ⸣ ana tar-s ṣa x [...]⸢ ⸣

8 [...] x ⸢ ša2 U⸣ 4.MEŠ ina 1.3 ME E ina 2 UŠ ina 2.12 ŠU⸢ 2 ⸣ ki-s ṣir AN x [...]⸢ ⸣

Translation

1 [...] ... [...]

2 [...] width? ... [...] ... the width, 0;12? the la[rge] side [...]

3 [...] ... the small side ... [...] 30? was carried ... [...]

4 [...] from the days ... You see the day when it passes the position of the crossing; 32[... you subtract from 1,0 ...]

5 [...] there remains 27 (error for 28). You add 28 to your day, and you see the daywhen it passes the position of the crossing [...]

6 [...] what remains: you add 0;9,30 to 0;10,50, you compute half of it and youmultiply it by 32 day[s, it is 5;25,20 ...]

7 [...] ... you let proceed ... nothing. 50. Opposite ... [...]

8 [...] ... of the days: in 1,3 rising to daylight, in 2,0 the station, in 2,12 the setting: thenode of ... [...]

Philological remarks

1) Only illegible traces are visible.

2) x UŠ , ⸢ ⸣ „..., the width“: the term UŠ, „width“, follows a damaged sign, most likely anumber. Only a final vertical wedge of that sign remains visible, compatible with 1-šu,„sixty“. 12? : only one vertical wedge is clearly visible after the 10, which would yield⸢ ⸣11, but they are separated by an anomalously large empty space. However, in that spaceare visible faint traces of another vertical wedge (Fig. S4), which prompts a tentativereading 12, to be interpreted as 0;12. The traces that follow pūtu(SAG.KI), „side“, arecompatible with rabītu(GAL), „large“. Hence this line may be part of a declaration of themeasures of a trapezoid of „width“ 60 (days) and „large side“ 0;12.

3) The term pūtu(SAG.KI) s ṣe ertuḫ (TUR-tu4), „small side“, should be preceded by thenumerical value of that side. The partly preserved sign might be a 50 or a 30 followed bya separator (:). E: replaces the previous reading PI, but the meaning remains unclear. 30⸢uz⸣-zab-bil: replaces the previous reading 10 x ⸢ ⸣ ŠU2 ZALAG2 GIBIL (2). The formuzzabbil is a preterite tense of the Dt or Dtn stem, or a perfect tense of the D stem ofzabālu, „to carry“. This verb is thus far attested in mathematical contexts only inconnection with brick computations (21), but the related verbs wabālu, „to carry“, andtabālu, to „carry away“, appear in mathematical texts with a meaning „to multiply“ and„to subtract“, respectively (4, 21). The present context is insufficient for determining the

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intended meaning of uzzabbil. IGI? 2.50?: the 50 may also be read as 20 followed by aseparator (:). The meaning of these signs is not clear.

4) [...] ina/ultu(TA) ūmē(U4.MEŠ.AM3) KI UD x [x], ⸢ ⸣ „[...] from the days ...“: themeaning of this fragmentary phrase is unclear. KI: most likely to be read itti, „with“, orqaqqaru, „position“, but no plausible interpretation could be found. UD x : the sign⸢ ⸣following UD begins with a vertical wedge. One might read tam-mar, „you see“, but thisdoes not result in a meaningful sentence, so the correct reading remains unclear. We areon firm ground from ⸢u4-mu , ⸣ „day“, onwards, because this phrase clearly belongs to partII of the trapezoid procedure. 3 2 : only the left wedge of the 2 is preserved before the⸢ ⸣break.

5) U⸢ 4 -⸣ ka, „your day“: replaces the previous reading x -⸢ ⸣ ka (2). etēqa(DIB-qa). „topass“: replaces the previous reading HAB? UD (2).

7) This line contains a procedure for Jupiter that turns out to be distinct from thetrapezoid procedure (see the commentary). ME E, „rising to daylight“, is the synodicphenomenon of acronychal rising; UŠ, „station“, here denotes the second station, andŠU2, „setting“, is the last appearance (2).

8) This line also contains a procedure that is distinct from the trapezoid procedure (seethe commentary).

Commentary

Lines 1-3 deal with a trapezoidal figure but they are difficult to interpret due to their badstate of preservation and the presence of signs and words for which no suitable translationcould be established, in spite of several improved readings. The end of line 2 and thebeginning of line 3 contain a declaration of the measures of the trapezoid that presumablybelongs to part I of the trapezoid procedure. Lines 4-6 are now understood for the firsttime as belonging to part II of the trapezoid procedure. However, they do not constituteits beginning, which implies that some portion of lines 2-3 must also belong to part II.Even though a horizontal ruling is not visible below lines 6 or 7, but below line 8, it isnow clear that the trapezoid procedure ends either at the end of line 6, or in line 7 (seebelow).

Line 5 begins with the phrase „there remains“. In Late Babylonian astronomical andmathematical texts this introduces the outcome of a preceding operation (2). Thisoutcome is nearly always repeated when it is passed on to an immediately followingcomputation (2). Here the result is passed on as 28, but the number preceding it looks like27. Hence this 27 is very likely a damaged 28 or a scribal error for 28. The phrase „thereremains“ occurs almost exclusively after subtractions (2), which strongly suggests thatthe 28 was obtained by subtracting 32 days, the number partly preserved at the end of line4, from 60 days, which can be restored in the following gap. In line 5 the 28 days are tobe added to „your day“, resulting in the „day when it (= Jupiter) passes the position of the

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crossing“, which confirms that 28 days is a value of tc, in agreement with theinterpretation of the trapezoid algorithm.

In line 6 the area of the second partial trapezoid is computed, very likely in order toverify the partition. The correctness of this interpretation is supported by an OldBabylonian mathematical tablet, UET 5, 858, on which the partition of a trapezoid ofdifferent dimensions is followed by exactly the same type of verification (5, 11).

Line 7 is very difficult to interpret. It is provisionally assumed here that the trapezoidprocedure ended in line 6, but some initial part of line 7 may constitute the end of thetrapezoid procedure. In line 8 we have definitely left the trapezoid procedure, since itdeals with a new topic, namely the duration of various intervals between Jupiter's synodicphenomena measured in days. 1,3 (=63) days is close to 60 days, attested as the timebetween the first station and the acronychal rising of Jupiter (2). 2,0 (=120) days is anattested value of the time between the first station and the second station (2). 2,12 (=132)days is an attested value of the time between second station and last appearance (2).

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Fig. S1.

Photograph of Text B (lines 1-12). Reproduced from (2).

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Fig. S2

Photograph of Text C (lines 20-24).

15

Fig. S3.

Photograph of Text D (lines 1-9).

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Fig. S4

Photograph of Text E (lines 1-8).

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18

References and Notes 1. O. Neugebauer, Astronomical Cuneiform Texts (Lund Humphries, London, 1955).

2. M. Ossendrijver, Babylonian Mathematical Astronomy: Procedure Texts (Springer, New York, 2012).

3. J. P. Britton, Studies in Babylonian lunar theory: Part III. The introduction of the uniform zodiac. Arch. Hist. Exact Sci. 64, 617–663 (2010). doi:10.1007/s00407-010-0064-z

4. J. Høyrup, Lengths, Widths, Surfaces. A Portrait of Old Babylonian Algebra and Its Kin (Springer, New York, 2002).

5. A. A. Vaiman, Shumero-Vavilonskaya matematika III-I tysyacheletiya do n. e. (Izdatel’stvo Vostochnoy Literatury, Moscow, 1961).

6. J. Friberg, Mathematik, sec. 5.4k, Partitioned trapezoids and trapezoid stripes, in Reallexikon der Assyriologie, D. O. Edzard, Ed. (De Gruyter, Berlin, 1990), vol. 7, pp. 561–563.

7. J. Friberg, A Remarkable Collection of Babylonian Mathematical Texts. Manuscripts in the Schøyen Collection: Cuneiform Texts I (Springer, New York, 2007).

8. Materials and methods are available as supplementary materials on Science Online.

9. P. J. Huber, Zur täglichen Bewegung des Jupiter nach babylonischen Texten. Z. Assyriol. 52, 265–303 (1957).

10. O. Neugebauer, Mathematische Keilschrifttexte, I–III (Springer, Berlin, 1935–1937).

11. J. Friberg, Mathematics at Ur in the Old Babylonian period. Rev. Assyriol. Archeol. Orient. 94, 97–188 (2000).

12. O. Pedersen, Early Physics and Astronomy. A Historical Introduction (Cambridge Univ. Press, Cambridge, 1974).

13. E. D. Sylla, The Oxford calculators, in The Cambridge History of Later Medieval Philosophy, N. Kretzmann, A. Kenny, J. Pinborg, Eds. (Cambridge Univ. Press, Cambridge, 1982), pp. 540–563.

14. T. Freeth, A. Jones, J. M. Steele, Y. Bitsakis, Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism. Nature 454, 614–617 (2008). doi:10.1038/nature07130 Medline

15. A. Jones, Astronomical Papyri from Oxyrhynchus (American Philosophical Society, Philadelphia, 1999).

16. E. Leichty, Catalogue of the Babylonian Tablets in the British Museum, vol. VI, Tablets from Sippar 1 (1986).

17. M. Dietrich, Die neubabylonischen Konjunktionen, in Lišān mithurti: Festschrift Wolfgang Freiher von Soden zum 19. VI. 1968 gewidmet, Alter Orient und Altes Testament, M. Dietrich, W. Röllig, Eds. (Butzon and Bercker, Kevelaer, Germany, 1969), pp. 1, pp. 65–99.

18. J. Hackl, Der subordinierte Satz in den spätbabylonischen Briefen, Alter Orient Altes und Testament 341 (Ugarit, Münster, Germany, 2007).

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19. W. von Soden, Grundriss der akkadischen Grammatik, Analecta Orientalia 33 (Pontifical Biblical Institute, Rome, 1995).

20. M. Ossendrijver, Evidence for an instrumental meaning of ina libbi. Nouvelles Assyriologiques Brèves et Utilitaries 44, 51 (2010).

21. O. Neugebauer, A. Sachs, Mathematical Cuneiform Texts, Americal Oriental Series 29 (American Oriental Society, New Haven, CT, 1945).